Oct 31 Multivariable Differentials

Multivariable Differentials

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Recall that in single variable, if $u = f(x)$, then $du = f'(x)dx$. We can extend this to multiple variables.
If $w = f(x,y)$,
$$dw = f_x(x,y)dx + f_y(x,y)dy = \frac{\partial{}w}{\partial{}x}dx + \frac{\partial{}w}{\partial{}y}dy$$
Similar to single variable differentials, this also finds a tangent line to the 3D curve expressed by the function and approximates $w + \Delta{}w$.